Transcript Generalized Hough Transform

Generalized
Hough
Transform
The Generalized Hough Transform
From Standard to Generalized
HT
1.Standard Hough Transform
requires parametric
representation for desired
curve
2.This idea is generalized in
the Generalized Hough
Transform
Example: Human Face
recognition
•
Is there some attribute of the structure of
the head that we can exploit to help
estimate pose estimation?
•
–
•
Is this attribute invariant under change in
pose?
Or
“Can we model how this attribute varies with
pose?”
Hough Transform in General
1. Technique to isolate curves of a
given shape in an image
2. Standard Hough Transform (HT)
uses parametric formulation of
curves
3. Generalized Hough Transform
(GHT) extends for arbitrary curves
Key Idea to improve
correlation by voting
1.
When we compute the correlation by voting, we
spend most of the time casting bad votes.
2. Idea is to use extra shape information (e.g.
gradients) to cast fewer votes:
1.
O(n) complexity: For each of O(n) points on the
boundary, cast O(1) votes.
General Hough Algorithm Idea
• 1. explicitly list points on shape
• 2. make table for all edge pixels for target
• 3. for each pixel store its position relative to some
reference point on the shape
– ‘if I’m pixel i on the boundary, the reference point is at ref[i]’
The Generalized Hough Transform
1.Technique to find arbitrary curves
in a given image
2.Parametric equation no longer
required
3.Look-up table used as transform
mechanism
4.Two phases:
1.R-Table Generation phase
2.Object Detection phase
The Generalized Hough Transform
1. Standard Techniques allow for invariance to
scale and rotation in the plane
2. In general, objects in the real world are 3dimensional
3. Hence a single silhouette provides no
invariance to pose (i.e. rotation out of the
plane).
4. No pose estimation.
5. This is generalized to Surface Normal
Hough Transform
Building the
R-Table
in GHT
GHT: Building the R-Table
1. We are given the shape we want to localize
2. We build a lookup table for this shape, called R-Table
It will replace the need for a parametric
equation in the transform stage
GHT: Building the R-Table
GHT: Building the R-Table
GHT:
Building
the R-Table
GHT: Building
the R-Table
Object
Localization in
the R-Table
in GHT
GHT: Object Localization
GHT: Object Localization
GHT: Object Localization
Conclusions
on
GHT
Conclusions on GHT
1. Standard Techniques allow for
invariance to scale and rotation in the
plane
2. In general, objects in the real world
are 3-dimensional
3. Hence a single silhuette provides no
invariance to pose (i.e. rotation out of
the plane).
4. No pose estimation.
5. Now show more details
Generalized Hough
Transform
Algorithm
Algorithm of the General
Hough Transform
Hough Transform for Curves
• The H.T. can be generalized to detect any
curve that can be expressed in parametric
form:
–
–
–
–
Y = f(x, a1,a2,…ap)
a1, a2, … ap are the parameters
The parameter space is p-dimensional
The accumulating array is LARGE!
Generalized Hough
Transform
algorithm
• Find all desired points in image
• For each feature point
– for each pixel i on target boundary
• get relative position of reference point from i
• add this offset to position of i
• increment that position in accumulator
• Find local maxima in accumulator
• Map maxima back to image to view
Generalizing the H.T.
The H.T. can be used even if the curve has not a
simple analytic form!
(xc,yc)
fi
ai
Pi
xc = xi + ricos(ai)
ri
yc = yi + risin(ai)
1. Pick a reference point (xc,yc)
2. For i = 1,…,n :
1. Draw segment to Pi on the boundary.
2. Measure its length ri, and its
orientation ai.
3. Write the coordinates of (xc,yc) as a
function of ri and ai
4. Record the gradient orientation fi at Pi.
3. Build a table with the data, indexed by fi .
Generalizing the H.T.
Suppose, there were m different gradient orientations:
(m <= n)
aj
rj
fj
(xc,yc)
ri afi
i
Pi
xc = xi + ricos(ai)
yc = yi + risin(ai)
f1
(r11,a11),(r12,a12),…,(r1n1,a1n1)
f2
(r21,a21),(r22,a12),…,(r2n2,a1n2)
.
.
.
.
.
.
fm
(rm1,am1),(rm2,am2),…,(rmnm,amnm)
H.T. table
Generalized H.T. Algorithm:
Finds a rotated, scaled, and translated version of the curve:
1.
Form an A accumulator array of possible
reference points (xc,yc), scaling factor S
q
and Rotation angle q.
2.
q
For each edge (x,y) in the image:
1.
Compute f(x,y)
2.
For each (r,a) corresponding to
f(x,y) do:
1.
q
xc = xi + ricos(ai)
yc = yi + risin(ai)
3.
For each S and q:
1.
xc = xi + r(f) S cos[a(f) + q]
2.
yc = yi + r(f) S sin[a(f) + q]
3.
A(xc,yc,S,q) = A(xc,yc,S,q) + 1
Find maxima of A.
Another variant of the Generalized
Hough Transform
Find Object Center
( xc , yc ) given edges ( xi , yi , fi )
A( xc , yc )
Create Accumulator Array
Initialize:
A( xc , yc )  0 ( xc , yc )
For each edge point
( xi , yi , fi )
For each entry
rki
in table, compute:
xc  xi  rki cos a ki
yc  yi  rki sin a ki
Increment Accumulator:
Find Local Maxima in
A( xc , yc )
A( xc , yc )  A( xc , yc )  1
Generalize HT applied for circuits
Properties of Generalized Hough Transform
• What can we do when the curve we want to detect is not easily described
parametrically?
1.
~ By this, we mean, it cannot be captured in a relatively small number of parameters.
2.
~ Recall, the dimensionality of the Hough space equal the number of parameters!
• The GHT constructs a parametric description of an arbitrary shape based on a learning
process.
• This parametric description is not, in general, compact.
• We will begin by assuming the size, shape, and rotation (orientation) of the region is
known a priori. (Or that we want only to detect instances of a given size and
orientation.
1.
~ The voting space is (equivalent to) image space, 2D,
and rotation.
in the case of known size
2.
~ We will see how to deal with unknown orientation and size shortly -- with a 4D
Hough space.
X R: An
arbitrary reference point inside the shape.
j
r : The length of the j-th line from the reference point to
the shape perimeter, intersecting at a point of tangent
angle ø.
f : The
angle of the (current) tangent(s) to the perimeter.
a j: The orientation of the j-th line segment.
The list of (
given f and
a partial characterization of the shape.
j
r ,j a ) pairs, for a
X Rconstitutes
• By sweeping the tangent angle (ø) over the range (0,2π) in
some reasonable quantization (!), we build what is called
the R-table (reference table) description of the shape.
f1 :
f2 :
fk :
(r1 , a 1 );
(r21 , a 1
2 );
(rk1 , a 1
k );
1
1
(r1 ,a 1 );
(r22 ,a 2
2 );
(rk2 ,a 2
k );
2
2
....
....
....
( r1 1 , a 1 1 );
2
(r2n 2 , a n
2 );
k
(rkn k , a n
k );
n
n
• Each pixel x (say, a detected edge point) with local
orientation ø provides evidence (votes for) reference
points at the set of locations indicated by the list in the
R-table for that tangent direction...
{ x1  r ( f ) cos[ a (f )], x 2  r ( f ) sin[ a ( f )]}
• A vote is cast for each (r ,
) pair in a
the list for that ø value.
The voting space is isomorphic to image space.
• Again, this assumes known size and orientation for all
appearances of the shape.
• After all the edge points have voted for all of their possible
reference points, we interrogate the voting space for
significant local maxima. These suggest possible
detections of the shape of interest.
• If we have not prenormalized for size (S) and rotation ( )
then our voting space is four dimensional and the reference location
receiving the vote(s) for a given edge point and R-table entry is:
x1R  x1  r(f )S cos[ a (f )   ]
x2R  x 2  r(f )S sin[a(f )   ]
• Now, we interrogate the 4D accumulator array to recover likely locations,
scale, and orientation for appearances of the shape.
• This is really a fancy form of a template match -- but one that is far more
robust than a straightforward template matching algorithm.
• Selecting among multiple possible shapes requires multiple R-tables,
multiple voting spaces.
• But, so does looking for lines and circles in the same image....
Generalized HT in biologically
motivated robotics
Bimodal Active Stereo
Many simultaneous problems in
robotics
Research Philosophy
The main concept of Radon
Transform
The main concept of Radon
Transform
Hough Transform: Comments
• Works on Disconnected Edges
• Relatively insensitive to occlusion
• Effective for simple shapes (lines, circles, etc)
• Trade-off between work in Image Space and Parameter Space
• Handling inaccurate edge locations:
• Increment Patch in Accumulator rather than a single point
H.T. Summary
• H.T. is a “voting” scheme
– points vote for a set of parameters describing a line or
curve.
• The more votes for a particular set
– the more evidence that the corresponding curve is
present in the image.
• Can detect MULTIPLE curves in one shot.
• Computational cost increases with the number of
parameters describing the curve.
end